Only if you have a poor or too small chimney void.
I have a rather larger one and the only time hat I have evr had smoke come out the front of the oven was when I was burning what I believed was dry red gum, I tried to encourage it with a petrol leaf blower on half throttle blasting the coals with 100km hr air.
Normally, even with a very scary fire, no smoke escapes out the front.
However, many builders have what I feel is too small and poorly designed chimney void and they have to live with the smoke stains in their oven's facade.
Do it right the first time and have no worries!

The advantage to building in a screened outdoor room is that you will have a LOT of chimney to get two feet above the roof. One of the main factors in chimney draw is how tall the chimney is. If you don't do something silly like trying to use undersized pipe, or use multiple bends your oven should draw like a champ.

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If you follow the advice above with adequate smoke chamber, I can imagine still getting a little whif of smoke out the front similar to lighting a fireplace. But get that 8 or 10 feet of flue the least bit warm and I doubt there will be anything else coming out the front. Some people even resort to lighting a sheet of newspaper and holding it up the flue to establish that first draft, but it seems that would cause as much smoke to come out the front as just lighting the fire and letting it go as is. Just start with a small fire for the first minute or two to get that heat going the right direction and you shouldn't have any problem.

I'm doing essentially the same thing with my build and, like you, have that little niggling concern about smoke in the room. Guess I'll have to wait and see.

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